Run a UPnP Media Server and you can access things right through the XMB. I have approached rejetto about implementing something like this and it is on the to-do for the future. The PS3 browser has minimal capabilities, it does come in handy when your too lazy to move over to your computer or open up your laptop while gaming though.
I have tested this myself and I cant find this issue you are having... Maybe i don't have the same setup with my HFS. I don't know what the BBC iPlayer is but I would assume its a flash based player, and may require a newer version of flash than what is in the current firmware. A PC will always be more flexible for the obvious reasons, but I have no problems using either my PS3 or 360 as media streaming/playing devices. I wish they would incorporate MKV support though, all my avi codecs are supported but these new mkv HR vids basically don't play.
I dont like to advertise other servers

cause lets face it, HFS is the best

. But Tversity is a pretty neat little media server. It also has the capability to re-encode unsupported formats on the fly for the connecting device as it streams. Of course you need a decent processor, it depends on the job. You will always lose quality with this conversion process, I have found anyway. It will also compress videos so they fit down the connection without too much buffering.
http://tversity.com/It will work with most devices that support the UPnP Media Server capability.